Math for LQG: Books & Resources for Graduate Level Understanding

[fando]

I'd like to acquire more than a casual understanding of LQG, and to do that I need the math first. My math background has been acquired ad-hoc while pursuing a degree in Astronomy. A smattering of group theory here, enough differential geometry and tensor gymnastics to read Gravitation (Thorne & Wheeler), your standard linear algebra for QM (Cohen & Tanoudji), and all the prereqs. If possible, I'd like to know what books a math major would use because the pick-it-up-as-you-go method of learning math has left me with a very unreliable foundation. I'm actually serious about this as I'm trying to decide what I'd like to do for grad school (I have a preference for GR and Astrophysics).

So what math topics would I need to study and what books/resources are recommended in order to understand LQG at the graduate level?

 

[marcus]

John Baez recently replied to a question like this on Usenet sci.physics.research ("spr"). It was someone who planned to do a Masters thesis on LQG and wanted to know what to read, and Baez listed some things.

I would suggest that you get on Usenet's "spr" and ask Baez this very question you posted here. Baez has been active in this field and is nice about answering questions.

Also, since your question is serious and ought to command their attention. Why not ask Carlo Rovelli and Abhay Ashtekar? Rovelli, I understand, is currently working on a graduate-level textbook on LQG. If Ashtekar does not reply to your email, try his postdoc at PSU, Martin Bojowald.

the best place for LQG appears to be Pennsylvania State University's Center for Gravitational Physics and Geometry.
That is where Ashtekar is. But Pittsburg must be OK too because
Rovelli is there.

If you are outside the US, please let me know and I can suggest people and centers in other countries. Send a PM (personal message) about this if you want.

As for books------you sound already well prepared, with differential geometry, group representations, and linear spaces.
(You did not mention measure theory and there is one important measure on a function space in the theory---also distribution theory----but these are linear spaces topics.) I would suggest that you print off a copy of a 52-page introduction
"Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance" arXiv: gr-qc/9910079, by Marcus Gaul and Carlo Rovelli, and go thru it and find out first hand what mathematics gaps are holding you up.
In other words, don't wait until you are sure you have all the mathematical preparations---instead, try the field out immediately and see just what you actually do need.

Several people here may have specific reading list suggestions.
If anything else occurs to me I will post it later.

Originally posted by fando
I'd like to acquire more than a casual understanding of LQG, and to do that I need the math first. My math background has been acquired ad-hoc while pursuing a degree in Astronomy. A smattering of group theory here, enough differential geometry and tensor gymnastics to read Gravitation (Thorne & Wheeler), your standard linear algebra for QM (Cohen & Tanoudji), and all the prereqs. If possible, I'd like to know what books a math major would use because the pick-it-up-as-you-go method of learning math has left me with a very unreliable foundation. I'm actually serious about this as I'm trying to decide what I'd like to do for grad school (I have a preference for GR and Astrophysics).

So what math topics would I need to study and what books/resources are recommended in order to understand LQG at the graduate level?

 

[fando]

Ok, thanks for the info! I'll pursue the leads--they're sufficient to get me started. (I'm in the US. My alma mater is Caltech, but I've been out of the academic loop for awhile.)

 

[marcus]

Originally posted by fando
Ok, thanks for the info! I'll pursue the leads--they're sufficient to get me started. (I'm in the US. My alma mater is Caltech, but I've been out of the academic loop for awhile.)

Hi fando, I notice you were still around (Imagine's thread about imaginary distance) and quoting "Gravitation" to the effect that it had to be "put to the sword" so I will continue this thread at a kind of low level in case its useful'

I will list my favorite LQG papers---a short list of mostly ones I think are very good. This carries no weight of authority but I might as well tell you what I personally have found the best written most informative.

Will type 'em in shortly unless something distracts me

 

[marcus]

fando,

About math "prerequisites" I'd suggest you look at some of these papers and see for yourself what math you would need to delve into them. I kind of suspect you have a lot of the necessaries already.

Cosmology is the ultimate test of a theory of spacetime and gravity IMHO and knowing some cosmology gives valuable perspective on LQG. So I am including Lineweaver's June 2003 cosmology tutorial even tho it nowhere mentions LQG.

1. Ashtekar "Quantum Geometry in Action: Big Bang and Black Holes" (math-ph/0202008)

2. Bojowald/Morales-Tecotl "Cosmological Applications of Loop Quantum Gravity" (gr-qc/0306008)

3. Rovelli "Loop Quantum Gravity" (LivingReviews 1998)

4. Lineweaver "Inflation and the Cosmic Microwave Background" (astro-ph/0305179)

5. Gaul/Rovelli "Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance" (gr-qc/9910179)


6. Baez "An introduction to spin foam models of BF theory and quantum gravity" (gr-qc/9905087)



---------the rest is footnotes-------

I want to mention this even tho I'm not sure I'd recommend it
Baez "Spin Foam Models" gr-qc/9709052

I like a 1998 popular article by Ashtekar very much "Quantum Mechanics of Geometry"
It is in the "links" section of the CGPG website (Center for Gravitational Physics and Geometry)
You can get to CGPG just by typing those 4 letters into google.

 

[marcus]

on the personal side, here is

http://www.rediff.com/news/1999/may/15us2.htm

a 4-page profile, with photo, of Abhay Ashtekar.
Turns out he was born around 1950 in Maharashtra
India, there is also a picture of one of his teachers
Roger Penrose.

Ashtekar's institute (CGPG) is producing 2-4 PhD's a year
and their dissertations are on-line with everything else
at the Center's site, so you can check out the quality of
the research of the people coming up.

"Links" at the CGPG site has what I think is a really good
popular article by Ashtekar written at "Scientific American"
level, called "Quantum Mechanics of Geometry".
The date is 1998. Found it on tip from wolram

http://cgpg.gravity.psu.edu/

 

[fando]

Great stuff! I think I have an idea of what rough edges need polishing. Thanks for the pointers and papers!